Finite Group Databasebeta

← #7231926

Subgroups — #7231926

Order 512.

1255 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#53125C212yesmaximal, complemented
2#33284C212yesmaximal, complemented
3#32857C211yesmaximal, characteristic, complemented
4#32814C211yesmaximal, characteristic
5#32252C214yesmaximal, complemented
6#31831C214yesmaximal, complemented
7#31665C214yesmaximal, complemented
8#29079C214yesmaximal, complemented
9#28902C214yesmaximal, complemented
10#27826C214yesmaximal, complemented
11#21677C211yesmaximal, characteristic, complemented
12#2832—24no—
13#2814—24no—
14#2702—24no—
15#2695—24no—
16#2647C2212yes—
17#2641C2212yes—
18#2563C2214yes—
19#2550C2214yes—
20#2513C2214yes—
21#2512C2214yes—
22#2452C2214yes—
23#2445C2214yes—
24#2402C2211yescharacteristic
25#2101C2214yes—
26#2101C2214yes—
27#2088C2214yes—
28#2084C2214yes—
29#2073C2214yes—
30#2065C2214yes—
31#2054C2214yes—
32#2054C2214yes—
33#2025C2214yes—
34#2023C2214yes—
35#2023C2214yes—
36#2011C2214yes—
37#1993C2214yes—
38#1992C2212yes—
39#1991C2212yes—
40#1988C2214yes—
41#1972C2214yes—
42#1961C2212yes—
43#1957C2212yes—
44#1940C2212yes—
45#1936C2214yes—
46#1936C2214yes—
47#1936C2214yes—
48#1934C2214yes—
49#1934C2214yes—
50#1926C2214yes—
51#1920C2214yes—
52#1917C2214yes—
53#1879C2214yes—
54#1879C2214yes—
55#1832C2211yescharacteristic
56#1804C2214yes—
57#1771C2214yes—
58#1770C2212yes—
59#1762C2214yes—
60#1726C2211yescharacteristic
61C22×D8C2314yes—
62C2×C4⋊Q8C2314yes—
63C2×#127C2312yes—
64C2×#126—24no—
65C2×#125C2312yes—
66C2×#123—24no—
67C2×C22⋊Q8—24no—
68C2×C4×D4C2314yes—
69C2×#106C2314yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2314yes—
72C2×#102C2314yes—
73C2×#102C2314yes—
74C2×#102C2314yes—
75C2×#102C2314yes—
76C2×#102C2314yes—
77C2×#102C2314yes—
78C2×#97C2312yes—
79C2×#97C2311yescharacteristic
80C2×#97C2314yes—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2312yes—
83#400—24no—
84#400D414yes—
85#398—24no—
86#398—24no—
87#398D414yes—
88#397—24no—
89#397—24no—
90#397—24no—
91#396—24no—
92#396—24no—
93#395—24no—
94#394—24no—
95#394—24no—
96#394—24no—
97#394—24no—
98#393—24no—
99#393—24no—
100#393—24no—
101#391—24no—
102#390C2314yes—
103#390C2314yes—
104#390C2314yes—
105#390C2314yes—
106#390C2314yes—
107#389—24no—
108#389C2314yes—
109#389C2314yes—
110#389C2314yes—
111#388—24no—
112#388—24no—
113#388—24no—
114#386—24no—
115#385—24no—
116#384C2312yes—
117#382C2312yes—
118#382C2314yes—
119#380—24no—
120#380—24no—
121C4×C4⋊C4C2314yes—
122C4×C4⋊C4C2314yes—
123C4×C4⋊C4C2312yes—
124#376C2311yescharacteristic
125#340C2314yes—
126#340C2314yes—
127#340C2314yes—
128#340C2314yes—
129#340C2314yes—
130#340C2314yes—
131#340C2314yes—
132#340C2314yes—
133#336C2314yes—
134#336C2314yes—
135#336C2314yes—
136C22×Q8—24no—
137C22×D4—24no—
138C22×D4C2414yes—
139C23×C4—24no—
140C2×D8—44no—
141C2×D8—44no—
142C2×D8—44no—
143C22×C8C2414yes—
144C22×C8C2411yescharacteristic
145C22×C8C2412yes—
146C22×C8C2411yescharacteristic
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149#127—42no—
150#127—42no—
151#126—88no—
152#125—42no—
153#125—42no—
154#123—88no—
155C22⋊Q8—88no—
156C4×D4—44no—
157C4×D4—44no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2414yes—
168C2×C4⋊C4C2414yes—
169C2×C4⋊C4C2414yes—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C42—24no—
182C2×C42C2414yes—
183C2×C42C2412yes—
184C2×C42C2411yescharacteristic
185#106—44no—
186C4⋊C8—42no—
187C4⋊C8—44no—
188#102—28no—
189#102—44no—
190#102—44no—
191#102—44no—
192#102—44no—
193#102—44no—
194#97—41no—
195#97—44no—
196#97—42no—
197C4×C8—22no—
198C4×C8—42no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95C2×D414yes—
207#95C2×D412yes—
208#95C2×D414yes—
209#95C2×D412yes—
210#95C2×D414yes—
211C24—24no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—84no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—84no—
222C2×D4—84no—
223C2×D4—84no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—48no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4C22×D414yes—
231C22×C4C22×D412yes—
232C22×C4C22×D414yes—
233C22×C4C22×D414yes—
234C22×C4C2511yescharacteristic
235D8—88no—
236C2×C8—22no—
237C2×C8—21no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—24no—
242C2×C8—22no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C82−1+412yes—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4—84no—
260C4⋊C4—84no—
261C22⋊C4—48no—
262C22⋊C4—84no—
263C22⋊C4—84no—
264C22⋊C4—84no—
265C22⋊C4—84no—
266C22⋊C4—84no—
267C22⋊C4—84no—
268C22⋊C4—84no—
269C22⋊C4—84no—
270C22⋊C4—84no—
271C22⋊C4—84no—
272C42—21no—
273C42—22no—
274C42—21no—
275C42—24no—
276C42—22no—
277C42—24no—
278C42—84no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23C23×D411yescharacteristic
287Q8—168no—
288D4—44no—
289D4—44no—
290D4—44no—
291D4—44no—
292D4—168no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—168no—
314C2×C4C2×2−1+411yescharacteristic
315C2×C4#58512yes—
316C2×C4#58512yes—
317C2×C4#58511yescharacteristic
318C8—22no—
319C8—41no—
320C8—42no—
321C8—44no—
322C22—84no—
323C22—84no—
324C22—84no—
325C22—84no—
326C22—84no—
327C22—84no—
328C22—84no—
329C22#349711yescharacteristic
330C22#350012yes—
331C22#358514yes—
332C4—44no—
333C4—44no—
334C4—44no—
335C4—42no—
336C4—84no—
337C4—84no—
338C4—84no—
339C4#353111yescharacteristic
340C4#355311yescharacteristic
341C4#356812yes—
342C2—164no—
343C2#4623911yescharacteristic
344C2#6238712yes—
345C2#6257014yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32814C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2402 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #53125C2semidirect
2. #33284C2semidirect
3. #32857C2semidirect
5. #32252C2semidirect
6. #31831C2semidirect
7. #31665C2semidirect
8. #29079C2semidirect
9. #28902C2semidirect
10. #27826C2semidirect
11. #21677C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#531252
#332842
#328572
#328142
#322522
#318312
#316652
#290792
#289022
#278262
#216772